Make the original mirrored object invisible
Don't know how to do this properly, but here is what I came up with:

You could create another object that envelops the object you want to hide and use a boolean modifier with difference set as the operation. Then just hide the enveloping object as shown in the image :-



You could offset the mirrored object's UVs (see image) by (1, 1), then in the shader substract (1, 1) from UV coordinates - the mirrored object will have original UVs and the original object will have negative coordinates. Enable alpha clipping in the material, make transparent everything that has UV coordinates less than 0 and the original object should be invisible. Similar results could achieved by comparing the coordinates of axis the object was mirrored along. The original object would still cast shadows though so beware